The exponential-logarithmic equivalence classes of surreal numbers

arXiv:1203.4538v1 [math.AC] 20 Mar 2012

The exponential-logarithmic equivalence classes of surreal numbers.

Salma Kuhlmann and Mickaël Matusinski March 21, 2012

1 Introduction.

In his monograph [Gon86], H. Gonshor showed that Conway’s real closed field of surreal numbers carries an exponential and logarithmic map. Subsequently, L. van den Dries and P. Ehrlich showed in [vdDE01] that it is a model of the elementary theory of the field of real numbers with the exponential function. In this paper, we give a complete description of the exponential equivalence classes (see Theorem3.4) in the spirit of the classical Archimedean and multiplicative equivalence classes (see Theorem 2.4and Proposition2.5). This description is made in terms of a recursive formula as well as a sign sequence formula for the family of representatives of minimal length of these exponential classes.

This result can be seen as a step towards proving that the field of surreal numbers No can be described as an exponential-logarithmic series fieldK^EL(for some subfield of generalized seriesK of No) and a field of transseries T. Indeed, we conjecture that our representatives of the exponential classes are the fundamental monomials of K - say the initial fundamental monomials - in the sense of [KM11], or the log- atomic elements (i.e. the monomials which remains monomials by taking iterated log’s) in the sense of [vdH06,Sch01]. Such a description would allow us to exploit these representations to introduce derivations on the surreals. Indeed, we know how to define derivations on exponential-logarithmic series fields [KM12,KM11] and on transseries fields [vdH97,Sch01].

In Section2, we give a concise summary of the recursive definitions of the field operations on No, as well as the definition of the exponential exp and logarithmic log maps, and generalized epsilon numbersǫNo. Our exposition is based on [Gon86].

Of particular interest to us is the analysis of certain equivalence relations on the sur- real numbers. Conway [Con01, p 31-32] introduced and studied theω-map to give a complete systemω^No(:=the image of No under this map) of representatives of the Archimedean additive equivalence relation. In [Gon86, Theorem 5.3], exploiting the convexity of the equivalence classes, Gonshor describes such a representativeω^aas the unique surreal of minimal length in a given class. By a simple modification of their arguments, we describe a complete systemω^ωNoof representatives of the Archimedean multiplicative equivalence relation. In Section3, we introduce and study what we call

1

theκ-map to give a complete systemκNo(:=the image of No under this map) of rep- resentatives of the exponential equivalence relation (Definition3.1and Theorem3.4).

We observe that:

ǫ_No(κ_No(ω^ωNo(ω^No(No.

Section4is devoted to establish the sign sequences of these representatives (see Theorem4.3). Then, in Section5, we introduce the notion of Transserial-Exp-Log fields, which unifies the notion of transseries and exp-log series. We conjecture that No is such a TEL field (Conjecture5.2).

We thank Joris van der Hoeven : we started with him the project of endowing NO with a derivation, and since then had several valuable exchanges on this topic in connection with the fields of surreal numbers and of transseries. This helped to lead us to the results we present here.

2 Preliminaries on surreal numbers.

2.1 Inductive definitions.

This section is dedicated to fix the notations and recall some of the definitions and results obtained in [Con01] and [Gon86]. We will also use results from [Ehr01] and [vdDE01].

We denote by On the proper class of ordinal numbers. A surreal number is any map a :α→ {⊖,⊕}whereα∈On. Any surreal may be canonically represented as a sequence of pluses and minuses called its sign sequence. The ordinalαon which the surreal number a is defined is called its length. Set l(a) :=α. The proper class of all surreal numbers is denoted by No.

It is totally ordered as follows. Let a and b be any two surreal numbers with l(a)≤ l(b). Consider the sign sequence of a completed with 0’s so that the two sign sequences have same length. Then consider the lexicographical order between them, denoted by

≤, based on the following relation:

⊖<0<⊕.

A subset of No is said to be initial if it is made of surreals whose lengths are less than or equal to a given ordinal. As in [Ehr01], No is endowed with a partial ordering called the simplicity :

a is simpler than b, write a<_sb⇔a is a proper initial segment of b.

Given a surreal a∈No, any surreal number a^′<_sa is called a truncation of a.

The construction of surreal numbers makes heavy use of the key idea of "cuts be- tween sets" in the classical construction of real numbers. Take two pairs (L,R) (as

"Left" and "Right") and (L^′,R^′) of subsets of No with L <R and L^′ <R^′. (L^′,R^′) is said to be cofinal in (L,R) if for any (a,b)∈L×R, there is (a^′,b^′)∈L^′×R^′, such that a<a^′<b^′<b. Note that cofinality is a transitive relation. The following results are fundamental. In particular, given a surreal a∈No, we will considere the pair (La,Ra) where La ={b∈No ; b<a and b<sa}and Ra={b∈No ; b>a and b<sa}.

Theorem 2.1 (Existence and cofinality) 1. [Gon86, Theorem 2.1] For any pair (L,R) with L <R, there exists a unique surreal a∈No of minimal length such that L<a<R. We sethL|Ri:=a∈No, and call it the cut between L and R.

2. [Gon86, Theorem 2.6] Suppose thathL|Ri=a∈No, L^′<a<R^′for some pair (L^′,R^′) cofinal in (L,R), thenhL^′|R^′i=a.

3. [Gon86, Theorems 2.8 and 2.9] One always has a=hL_a|Raiand, for any cut hL^′|R^′i=a, then (L^′,R^′) is cofinal in (La,Ra).

Definition 2.2 This representation a=hLa |Raiof a is called its canonical cut. By abuse of notation, we also denote the canonical cut by a=ha^L|a^Riwhere a^Land a^R are general elements of the canonical sets Laand Ra(e.g. a^L=n if La=N).

The preceding results allow to define maps and operations on surreal numbers re- cursively, that is by induction on the length of the surreals considered since a^Land a^R are simpler than a. Concerning the algebraic structure of No, we have:

Theorem 2.3 1. [Gon86, Theorems 3.3, 3.6, 3.7 and Ch. 5, Sect. D] The proper class No endowed with its lexicographical ordering≤and the following opera- tions, is a real closed field (in the sense of proper classes): for any a,b∈No,

addition: a+b := ha^L+b,a+b^L|a^R+b, a+b^Ri neutral element : 0 = h∅ | ∅i;

multiplication: a.b := ha^L.b+a.b^L−a^L.b^L, a^R.b+a.b^R−a^R.b^R| a^L.b+a.b^R−a^L.b^R, a^R.b+a.b^L−a^R.b^Li neutral element : 1 = ⊕=h0| ∅i.

2. [Ehr01, Theorems 9 and 19] Any divisible ordered abelian group, respectively any real closed field, is isomorphic to an initial subgroup of (No,+), respectively an initial subfield of (No,+, .).

Note that Gonshor proves also, for the cited operations as well as for the following maps, the so-called uniformity properties [Gon86, Theorems 3.2, 3.5 etc.]. For ex- ample in the case of the addition, this means that a+b may be obtained by taking any cuts a=hL|Riand b=hL^′|R^′iinstead of taking their canonical cuts, and applying the same formula as above. In other words, the formulas do not depend on the cuts for which a^L, a^R, b^L, b^Rare general elements.

Any real number is identified to the sign sequence corresponding to its binary expansion. Thus they are the sequences of length finite or equal toω, the later being non ultimately constant. These sequences form a subfield of the surreals having the least upper bound property, i.e. a copy of the ordered set of the reals:R⊂No.

Surreal numbers a∈No such that a >Ror a<Rare called infinitely large: we denote by No^≫1 their proper class. Those that verify 0<a<R_>0or 0>a>R_<0are called infinitesimals: we denote by No^≪1their proper class.

Any ordinal numberα ∈ On is identified to the surreal number whose sign se- quence is a sequence of⊕’s of lengthα. The operations precedingly defined on the

surreal numbers correspond to the natural sum and product (see e.g. [Hau44]). So we consider On⊂No.

In the following theorem, we sum up the main results about the so-called Conway normal form of a surreal number:

Theorem 2.4 (Conway normal form) 1. [Gon86, Theorems 5.1 to 5.4] The re- cursive formula:

∀a∈No,ω^a:=h0, n.ω^aL|ω^aR/2ⁿ} (where it is understood that n∈N), defines a map:

Ω: No → No a 7→ Ω(a) :=ω^a

with values in No_>0and that extends the exponentiation with baseωof the ordi- nals. Moreover, for any a∈No,ω^ais the representative of minimal length of its Archimedean equivalence class (∀x,y∈No>0, x∼arch y⇔ ∃n∈N, n.x≥y≥ x/n).

2. [Gon86, Theorems 5.5 to 5.8] The field of surreals is a field of generalized series [KM12, Definition 2.5] in the following sense: any surreal number a∈No can be written uniquely as a = X

i<λ

ω^air_i where the transfinite sequence (a_i)_i<λ is strictly decreasing and, for any i, ri ∈ R\ {0}. In particular,ω^Nois seen as the group of (generalized) monomials:

No=R((ω^No)).

In particular, the proper class ω^Nois a complete system of representatives of the Archimedean equivalence classes of No_>0, each of its elements being the reprensenta- tive of minimal length in its class.

Moreover, note that the map denoted by Ind in [Gon86], which sends any surreal a to the exponent a0of the leading monomial of its normal form, is a valuation. It is the natural valuation of the real closed field No [Kuh00]. In particular, Ind(ω^a)=a for any a∈No.

We deduce immediately that No is a Hahn field of series [Hah07] in the following sense:

Proposition 2.5 1. For any a ∈ No,ω^ωa is the representative of minimal length in its equivalence class of comparability (∀x,y ∈ No^≫1_>0, x ∼comp y ⇔ (∃n ∈ N, xⁿ≥y≥x^1/n, and we set 1/x∼comp x).

2. Any surreal a ∈ No can be written uniquely a = X

i<λ









 Y

j<λ_i

ω^ωbi,jsi,j











ri where, for any i, we identifyY

j<λi

ω^ωbi,jsi,j

=ω^ai where ai =X

j∈λi

ω^bi,j (so the tranfinite

sequences (ai)i<λand for any i, (bi,j)j<λ_i, are strictly decreasing, and for any i,j, si,j,ri∈R\{0}). In particular, we callω^ωNothe chain of fundamental monomials of No [KM12, Definition 2.2]:

No=R

ω^ωNoR .

The proper class ω^ωNo is a complete set of representatives of the comparability classes of No^≫1_>0, each of its elements being the one of minimal length in its class.

Following the ideas of Conway, Gonshor described also a proper class of general- ized epsilon numbers [Con01, p.35]:

Theorem 2.6 The recursive formula:

∀a∈No,ǫa:=hΩⁿ(1), Ωⁿ(ǫa^L+1)|Ωⁿ(ǫa^R−1)}

(where it is understood that n∈N), defines a map:

ǫ: No → No

a 7→ ǫ(a) :=ǫ_a

with values in No^≫1_>0 bigger than Ωⁿ(1) for any n, and which extends the classical epsilon numbers map on ordinals [Sie65]. Moreover,ǫ(No) is the proper class of all the fixed points of the mapΩ:

∀a∈No, ω^ǫa=ǫa.

Gonshor defines inductively in [Gon86] an exponential map exp : (No,+) → (No>0, .) which is surjective (and consequently its inverse, the logarithmic map log : (No>0, .)→(No,+)) using the Taylor expansion of the real exponential map. Here we give only the results we will need in this article:

Theorem 2.7 The exponential map exp : No→ No>0and its inverse log : No>0 → No, coincide on the real numbers with the usual real exponential and logarithmic maps, and are such that:

1.for any a∈No, exp(X

i<λ

ω^airi) = ω^{Pi<λωg(ai)ri}

where g(a) := hInd(a), g(a^L)|g(a^R)i;

2.for any b∈No, log(Y

j<λ

ω^{ωb js}j

) = X

j<λ

ω^h(bj)

where h(b) := h0, h(b^L)|h(b^R), ω^b/2ⁿi.

We have exp=log⁻¹and h=g⁻¹on No.

Note that g is not defined at 0.

2.2 Sign sequences.

Gonshor obtains in [Gon86] detailed results on the sign sequences of the surreals under the various operations and maps. We will use repeatedly his results. Moreover, we introduce in this section some new operations on the sign sequences.

Definition 2.8 • Given two surreal numbers a,b ∈ No, we define their concate- nation aNb as the juxtaposition of their sign sequences. We note that l(aNb)= l(a)+l(b), theordinal sumof l(a) and l(b).

• As in [Gon86, Theorem 9.5], for any surreal number a ∈ No, we can write its sign sequence as the following transfinite concatenation:

a=α0⊕Nβ0⊖Nα1⊕Nβ1⊖N· · ·

where for anyµ ∈ On used in this writing of a, we haveα_µ, β_µ ∈ On with in particularα_µpossibly 0 forµ=0 or for anylimit ordinalµ.

• Given a surreal number a∈ No, we denote by a⁺the total number of pluses in the sign sequence of a (a⁺∈No). Therefore is the ordinal sum of the packages of⊕’s:

a⁺=X

µ

α_µ=N_µ(α_µ⊕).

• Given a positive surreal number a∈No>0, we derive from it the surreal number

♭a obtained by suppressing the first⊕in the sign sequence of a.

Given a negative surreal number a∈No_<0, we derive from it the surreal number

♯a obtained by suppressing the first⊖in the sign sequence of a.

• Given a surreal number in normal formX

i<λ

ω^ai.r_i, we define its corresponding reduced sequence (a⁰_i)i<λ as follows. For any i < λ, a⁰_i is derived from aiby suppressing in its sign sequence the following⊖’s:

– given an ordinal numberν∈On, if ai(ν)=⊖and if there exists j<i such that aj(ξ)=ai(ξ) for anyξ≤ν, then suppress theνth⊖;

– if i is a successor ordinal, ai−1N⊖is a truncation of ai and if ri−1 is not a dyadic rationnal number, then suppress the⊖coming after ai−1 in the writing of ai.

Concerning the exponentiation with baseω, we recall [Gon86, Theorem 5.12 and Corollary 5.1]:

Theorem 2.9 • Given a surreal number a=α₀⊕Nβ₀⊖Nα₁⊕Nβ₁⊖N· · ·, for any ordinalµ∈On intervening in the writing of a, we setγ_µ:=X

λ≤µ

α_λ(ordinal sum). Then the sign sequence ofω^ais:

ω^a=ω^γ0⊕Nω^γ0+1β0⊖Nω^γ1⊕Nω^γ1+1β1⊖N· · ·

• Given a positive real number r=ρ0⊕Nσ0⊖N· · ·, the sign sequence ofω^a.r is ω^a.r=ω^aNω^a+♭ρ0⊕Nω^a+σ0⊖Nω^a+ρ1⊕Nω^a+σ1N· · ·

(ordinal multiplication). If r is negative, reverse all the signs in the preceding sequence.

• Given a surreal number in normal formX

i<λ

ω^ai.r_i, its sign sequence is:

N_i<λω^a0i.r_i where (a⁰_i)_i<λis the corresponding reduced sequence.

2.3 Generalized epsilon numbers

We recall [Gon86, Theorems 9.5 and 9.6]:

Theorem 2.10 • A surreal number a=α0⊕Nβ0⊖Nα1⊕Nβ1⊖N· · · is an epsilon number if and only ifα0,0, allαµdifferent from 0 are ordinary epsilon numbers satisfyingα_µ >l.u.b.{α_λ|λ < µ}and furthermoreβ_µis a multiple ofω^αµω for αµ,0 and a multiple ofω^γµωwhereδµ:=X

λ<µ

αλ(ordinal sum) forαµ=0.

• Letγ_µ:=X

λ≤µ

α_λ(ordinal sum). Then theµth block of pluses inǫaconsists ofǫ_γµ pluses and theµth block of minuses of (ǫ_γµ)^ωβ_µminuses.

3 The kappa map.

In this section we define and describe a new mapκ, which takes naturally place between the Conway-Gonshor mapsΩandǫ(see Remark4.5). As in [Kuh00, Remark 3.20], we introduce the notion of exponential equivalence relation for surreal numbers. Set expⁿand logⁿfor the nth iterate of the corresponding maps.

Definition 3.1 We set :

• the exponential equivalence relation to be:

∀x,y∈NO^≫1, x∼expy⇔ ∃n∈N, logⁿ(|x|)≤ |y| ≤expⁿ(|x|);

• the exponential comparison relation to be:

∀x,y∈NO^≫1, x≫_expy⇔ ∀n, logⁿ(|x|)>|y|.

Theorem 3.2 The recursive formula

∀a∈No, κ(a)=κ_a:=hexpⁿ(0), expⁿ(κ_aL)| logⁿ(κ_aR)i (where it is understood that n∈N) defines a map

κ: No → No

a 7→ κ(a) :=κ_a with values in No^≫1_>0 and such that:

(i) for any a,b∈No, a<b⇒κa≪expκb;

(ii) there is a uniformity property for this formula (i.e. the recursive formula does not depend on the choice of the cut for a).

Proof. We proceed by transfinite induction on the length of the surreals considered.

For l(a)=0, i.e. a=0=h∅|∅i, we have:

κ0 := hexpⁿ(0)| ∅i

= hn| ∅iby cofinality

= ω.

We consider a ∈ No with l(a) > 0, and suppose that the theorem holds for any b ∈ No with l(b)<l(a). Consider a canonical representation a = ha^L|a^Riof a. By the induction hypothesis, sinceκ_aL > Randκ_aL ≪_exp κ_aR, for any n ∈ N we have expⁿ(0) < expⁿ(κ_aL) < logⁿ(κ_aR). Thus the recursive formula for κa is well defined.

Moreover, since expⁿ(0) for n ∈ Nis cofinal in Rand is always part of the lower elements in the recursive formula, we haveκa >R, meaning thatκa ∈ No^≫1_>0 for any a∈No.

(i) To show the property a<b⇒κa≪expκb, we proceed as in [Gon86, Theorems 3.1, 3.4 et 5.2] by induction on the natural sum of l(a) and l(b). So we consider a<b in No. We denote by c the common initial segment of a and b. There are two cases. Either c=a or b: in this case, the property follows directly from the recursive definition ofκ, since a=b^Lor b =a^Rrespectively. Or we have a <c< b with c<_s a and c <_s b.

Then the natural sum of a and c, and of b and c, is less than the one of a and b. By the induction hypothesis, we obtain thatκ_a≪expκ_candκ_c<_expκ_b, soκ_a ≪expκ_b.

(ii) Concerning the uniformity property, we refer to [Gon86, Theorems 3.2, 3.5, Corollary to Theorem 5.2 etc]. Let a = hL |Ri ∈ No. By Theorem2.1(3.), (L,R) is cofinal in the canonical cut (L_a,R_a) of a. By the preceding property, it implies that (κ(L), κ(R)) and (κ(L_a), κ(R_a)) are cofinals, and thatκ(L)≪_exp κ_a ≪_exp κ(R)⇒κ(L)<

κ_a < κ(R) (these inequalities concern elements in No^≫1_>0). So we can apply Theorem 2.1(2.) to obtainκa=hκ(La)|κ(Ra)i=hκ(L)|κ(R)i.

Example 3.3 We compute some particular values:

κ₁ := hexpⁿ(0),expⁿ(ω)| ∅i

= hΩⁿ(ω)| ∅iby cofinality

= ǫ0;

κ−1 := hexpⁿ(0),| logⁿ(ω)i

= hn|ω^ω−niby cofinality

= ω^ω−ω ;

Theorem 3.4 A surreal number b∈No is of the formκafor some a∈No if and only if b is the representative of minimal length in an exponential equivalence class.

Proof. We got inspired by [Gon86, Theorem 5.3].

Consider a,b∈No such that b∼expκa ⇔ ∃n∈N, logⁿ(x)≤y≤expⁿ(x). So we have expⁿ(0)≪exp b and expⁿ(κ_aL)≪exp b≪exp logⁿ(κ_aR) for any n∈N. Soκais an initial part of b.

Conversely, we show by transfinite induction on l(b)∈On that for any b∈No^≫1_>0, there exists a∈No such that b∼expκa. First, by (ii) of Theorem3.2, we note that a is unique whenever it exists.

One hasω=κ0.

Consider b= hL_b|Rbi ∈ No^≫1_>0 with l(b) > ωand suppose that for any c∈ No with l(c) < l(b) the desired property holds. So, for any c ∈ Lb∪Rb, we have c ∼_exp κ_d for some d ∈No. Let L :={d ∈ No| ∃c∈ Lb, c∼_exp κ_d}and R :={d ∈No| ∃c∈ R_b, c∼_expκ_d}. Suppose that L∩R,∅. We consider d∈L∩R. So there exist c₁∈L_b and c₂ ∈ R_b such that c₁ ∼_exp κ_d and c₂ ∼_exp κ_d. Since c₁ <c < c₂, it follows that c∼_expκ_d.

Suppose now that L∩R=∅. We have L<R (if not we would have d1>d2⇒κd₁ ≫exp

κd₂ ⇒ c1 >c2 with c1 ∈Lband c2 ∈Rb). Note that, by definition,κ(L), respectively κ(R), is a complete set of representatives of the exponential-logarithmic equivalence classes containing the elements of Lb \ {0}, respectively of Rb. There are 3 different cases:

•either there are a∈L and n∈Nsuch that expⁿ(κa)≥b. Consider c1 ∈Lbsuch that c1∼expκa. Thus c1<b≤expⁿ(κa) and c1 ∼expκa∼expexpⁿ(κd). So b∼expκa;

•either there are a∈R and n∈Nsuch that logⁿ(κa)≤b. Simingly, we obtain b∼expκa

;

•or for any d1 ∈ L, d2 ∈ R and n∈N, we have expⁿ(κ_d1)<b <logⁿ(κ_d2). Consider c₁ ∈ L_b\ {0}. There is d₁ ∈ L such that c₁ ∼_exp κ_d1. In particular, c₁ ≤ expⁿ(κ_d1) for some n. As before, for any c₂ ∈ R_b, there exists d₂ ∈ R such that c₂ ≥log^m(κ_d2).

Theorem2.1(2.) applies, so b = hexpⁿ(0),expⁿ(κ_d1)| logⁿ(κ_d2)i = κ_a where we set

a :=hL|Ri.

4 Sign sequences formulae for the kappa map.

For any a∈No and n∈N, we denote byκa,n :=logⁿ(κa) andκa,−n=expⁿ(κa), where log⁰and exp⁰are equal to the identity map on No. The aim of this section is to show thatκa,n andκa,−nfor any n are elements ofω^ωNo, and to give their sign sequence. In order to do so, we introduce the following auxiliary mapιand give its sign sequence.

Notation 4.1 • Consider a surreal number a =α0⊕Nβ0⊖Nα1⊕Nβ1⊖N· · ·. For any ordinalµ∈On intervening in the writing of a, we set:

γ_µ:=X

λ≤µ

α_λ (ordinal sum).

• We setΩ⁰to be the constant map equal to 1 in No andǫ♭0:=0.

Lemme 4.2 The recursive formula:

∀a∈No, ι(a)=ι_a:=hι_aLNΩⁿ(ǫ_♭(aL)⁺+1)⊕ |ι_aRNn⊖i.

(where it is understood that n∈N) defines a map

ι: No → No

a 7→ ι(a) :=ι_a

such that for any a=α0⊕Nβ0⊖Nα1⊕Nβ1⊖N· · ·, the sign sequence ofιais given by:

ι_a=ǫ_♭γ0⊕Nωβ₀⊖Nǫ_♭γ1⊕Nωβ₁⊖N· · ·. A uniformity property holds for this map.

Proof. We proceed by transfinite induction on l(a)∈On. We haveι0=h∅ | ∅i=0.

Given a surreal number a∈ No with l(a)>0, we suppose that for any b with l(b) <

l(a),ιb is well defined, with the corresponding sign sequence. Consider a canonical representation a =ha^L |a^Riof a. For instance, in the case where l(a) is a successor ordinal and a=a^LN⊕. So a^R<sa^Let a^R >a^L. More precisely, either a^Lis only made of⊕’s in which case a^R =∅(case (i)), or a^Ris a truncation of a^L, ending with a group of⊖’s of a^Lof smaller number than the corresponding one in a^L(case (ii)). In the case (i), the sign sequence formula ofι_ais:

ιa = hι_aLNΩⁿ(ǫ_♭(aL)⁺+1)| ∅i

= hǫ_♭(aL)⁺NΩⁿ(ǫ_♭(aL)⁺+1)| ∅i

= ǫ_♭(aL)⁺+1

= ǫ_♭a+

In the case (ii), denote by a^L=α0⊕Nβ0⊖N· · ·Nαµ⊕Nβµ⊖N· · ·. So a^R=α0⊕Nβ0⊖ N· · ·Nαµ⊕N˜βµ⊖with ˜βµ< βµ(in On) for someµ∈On. By the induction hypothesis, ι_aL =ǫ_♭γ0⊕Nωβ₀⊖N· · ·Nǫ_♭γµ⊕Nωβ_µ⊖N· · ·< ǫ_♭γ0⊕Nωβ₀⊖N· · ·Nǫ_♭γµ⊕Nωβ˜_µ⊖=ι_aR. So we obtain as desiredι_aLNΩⁿ(ǫ_♭(aL)⁺+1)< ι_aRNn⊖for any n∈N. Moreover,

ιa = hι_aLNΩⁿ(ǫ_♭(aL)⁺+1)|ι_aRNn⊖i

= h(ǫ_♭γ0⊕Nωβ₀⊖N· · ·Nǫ_♭γµ⊕Nωβ_µ⊖N· · ·)NΩⁿ(ǫ_♭(aL)⁺+1)| (ǫ_♭γ0⊕Nωβ₀⊖N· · ·Nǫ_♭γµ⊕Nωβ˜_µ⊖)Nn⊖i

= ǫ_♭γ0⊕Nωβ₀⊖N· · ·Nǫ_♭γµ⊕Nωβ_µ⊖N· · ·)Nǫ_♭(aL)⁺+1

= ǫ_♭γ0⊕Nωβ₀⊖N· · ·Nǫ_♭γµ⊕Nωβ_µ⊖N· · ·)Nǫ_♭(a)+.

The other cases for l(a) and a are analogous. They are left to the reader as an exercise.

Concerning the uniformity property, by the sign sequence formula forι, we note that for any surreal numbers a,b ∈ No, a < b ⇒ ιa < ιb. So the uniformity property follows from Theorem2.1in the same way as in [Gon86, Theorem 3.2, 3.5, Corollary

au Theorem 5.2] and in Theorem3.2.

The main result of this section is:

Theorem 4.3 1. For any a ∈ No, n ∈ N, we haveκa,n, κa,−n ∈ ω^ωNo. More pre- cisely:

κ_a,n = ω^ωb with b=ιaNn⊖;

κa,−n−1 = ω^ωb with b=ιaNΩⁿ(ǫ♭a⁺+1)⊕. In particular,κa=ω^ωιa.

2. With the Notation4.1, for any a∈No, n∈N, we have:

κa = ω^ωǫ♭γ0 ⊕N[(ω^ωǫ♭γ0ω)²β0]⊖Nǫ♭γ₁⊕N[(ǫ♭γ₁ω)²β1]⊖N· · · ; κ_a,n = κ_aN(ǫ_♭a+ωn)⊖;

κ_a,−n−1 = κ_aN(Ωⁿ⁺²(ǫ_♭a++1)⊕.

The second point of the theorem follows directly from the first one and the sign sequences for the mapsΩ(Theorem2.9) andι (Lemma4.2). The first point will be deduced from the following lemma.

Lemme 4.4 For any a∈No and anyλ∈On, if we setλ=ǫ_ν+µwhereǫ_νis the unique epsilon number (possibly 0) such thatǫ_ν< λ≤ǫ_ν+1, we have:

h(ιaNλ⊖) = ω^{ιaN(λ+1)⊖};

h(ιaNλ⊕) = ω^ιaNǫ⊕N♭µ⊕ avecλ >0.

Proof. We proceed by transfinite induction on (l(a), λ)∈−−−−−−−−→

On×On (the lexicographi- cal product of On with itself).

Suppose that l(a)=0 i.e. a=0. By the definition of the map h (see Theorem2.7(2.)), we have h(0)=h0|1/2ⁿi=ω⁻¹.

Moreover, 1=⊕=h0| ∅i. So:

h(⊕) = h0,h(0)|ω/2ⁿi

= hω⁻¹|ω/2ⁿi

= h⊕Nω⊖ |ω⊕N⊖iby the Theorem2.9

= ⊕=ω⁰.

Consider now (a, λ) ∈ No ×On with λ > 0 in On. Suppose that the lemma holds for any (b, µ) ∈ No×On with (l(b), µ) <lex (l(a), λ) in −−−−−−−−→

On×On. We have ιaNλ⊖ = h(ιa)^L |ιaNµ⊖iwithµ < λ in On. So, by the definition of h and by the induction hypothesis:

h(ιaNλ⊖) = h0, h((ιa)^L)|h(ιaNµ⊖), ω^ιaNλ⊖/2ⁿi

= h0, h(ι_aLNΩⁿ(ǫ_♭(aL)⁺+1)⊕)|ω^{ιaN(µ+1)⊖}, ω^ιaNλ⊖/2ⁿi(Lemma4.2)

= h0, ω^ιaLNΩⁿ(ǫ_♭(aL)⁺+1)⊕ |ω^ιaNλ⊖N(ǫ_♭a+n)⊖iby cofinality

= ω^ιaNλ⊖N(ǫ_♭a+ω)⊖

= ω^{ιaN(λ+1)⊖}by Theorem2.9.

We setλ = ǫν+µwhereǫν < λ ≤ ǫν+1. We have ιaNλ⊕ = ιaNǫν ⊕Nµ⊕ = hιaNǫν⊕Nρ⊕ |(ιa)^Riwithρ < µin On. So, by the definition of h and by the induction hypothesis:

h(ι_aNλ⊕) = h0, h(ι_aNǫ_ν⊕Nρ⊕)|h((ι_a)^R), ω^ιaNλ⊕/2ⁿi

= h0, ω^ιaNǫ_ν⊕N♭ρ⊕ |h(ι_aRNn⊖), ω^ιaNλ⊕/2ⁿi

= h0, ω^ιaNǫ_ν⊕N♭ρ⊕ |ω^{ιaRN(n+1)⊖}, ω^ιaNλ⊕/2ⁿi (note thatι_aRN(n+1)⊖> ι_aNλ⊕)

= h0, ω^ιaNǫ_ν⊕N♭ρ⊕ |ω^{ιaRN(n+1)⊖}, ω^ιaN(ǫ_♭a+ω^λ)⊕N(ǫ_♭a+ω^λn)⊖i (Theorem2.9and cofinality)

= ω^ιaNǫ⊕N♭µ⊕.

Consider now a ∈ No with l(a) > 0, and suppose that the lemma holds for any (b, µ)∈No×On with l(b)<l(a) in On. We haveιa=hι_aLNΩⁿ(ǫ_♭(aL)⁺+1)⊕ |ι_aRNn⊖i by Lemma4.2. So, by the definition of h and the induction hypothesis, we have:

h(ιa) = h0, h((ιa)^L)|h((ιa)^R), ω^ιa/2ⁿi

= h0, h(ι_aLNΩⁿ(ǫ_♭(aL)⁺+1)⊕)|h(ι_aRNn⊖), ω^ιa/2ⁿi(Lemma4.2)

= h0, ω^ιaLNΩⁿ(ǫ_♭(aL)⁺+1)⊕ |ω^{ιaRN(n+1)⊖}, ω^ιaN(ǫ_♭a+n)⊖i(Theorem2.9)

= h0, ω^ιaLNΩⁿ(ǫ_♭(aL)⁺+1)⊕ |ω^ιaN(ǫ_♭a+n)⊖iby cofinality

= ω^ιaN(ǫ_♭a+ω)⊖

= ω^ιaN⊖(Theorem2.9).

Proof of Theorem4.3(1.). We proceed by tranfinite induction on (l(a),n)∈−−−−−−→

On×N.

Suppose that l(a)=0, i.e. a=0. In the proof of the Theorem3.2, we proved that κ0=ω. Soκ0=ω^ω0as desired.

Let n ∈ N^∗. We suppose that the property holds for any (b,m) ∈ No×Nwith (l(b),m)<(l(a),n). We have:

κ_a,n = logⁿ(κ_a)

= log(κ_a,n−1)

= log(ω^ωb) with b=ιaN(n−1)⊖

= ω^{h(ιaN(n−1)⊖)}(Theorem2.7)

= ω^ωb with b=ιaNn⊖ (Lemma4.4) We also have:

κa,−1 = exp(κa)

= exp(ω^ωιa)

= ω^ωb with b=g(ω^ιa) (Theorem2.7)

= ω^ωb with b=ιaN⊕

(Lemma4.4: g(ω^ιa)=ιaN⊕ ⇔ω^ιa =h(ιa) and

κ_a,−n−1 = expⁿ⁺¹(κa)

= exp(κ_a,−n)

= exp(ω^ωb) with b=ιaNΩⁿ⁻¹(ǫ♭a⁺+1)⊕

= ω^ωb with b=g(ω^{ιaNΩn−1(ǫ♭a++1)⊕}) (Theorem2.7)

= ω^ωb with b=ιaNΩⁿ(ǫ_♭a++1)⊕ (Lemma4.4)

Consider a∈No with l(a)>0, and suppose that the property holds for any (b,m)∈ No×Nwith l(b)<l(a) and any m ∈N. By the definition ofκ(Theorem3.2) and by the induction hypothesis, we have:

κa = hexpⁿ(0), expⁿ(κ_aL)| logⁿ(κ_aR)i

= hn, ω^ωb|ω^ωc, ω^ιa/2ⁿi with b=ιa^LNΩⁿ(ǫ_♭(a^L)⁺+1)⊕) and c=ιa^RNn⊖

= ω^ωιa(by cofinality and definition ofι, Lemma4.2)

This finishes the proof of Theorem4.3.

Remark 4.5 1. One can find in [Gon86] partial results about these sign sequences.

Indeed, for any n∈N, any ordinalλ∈On, any epsilon-numberǫ_µwithµ∈On, we have:

logⁿ(ω^ωλ⊖) = ω^ω(λ+n)⊖by [Gon86, Theorem 10.15]

expⁿ(ω) = Ωⁿ⁺¹(1) by [Gon86, Theorem 10.14]

expⁿ(ǫµ) = Ωⁿ⁺¹(ǫµ+1) by [Gon86, Theorem 10.14]

.

2. By the Theorems2.9,2.10and4.3, we have:

ǫ_No(κ_No(ω^ωNo (ω^No(No.

5 Transseries and exp-log series.

Fields of Transseries [É92,vdH97,vdH06] or log-exp series [vdDMM97], and fields of exp-log series [Kuh00,KS05] are important non standard models of the theory ofR_exp, which is known to be o-minimal and model complete [Wil96]. Moreover, these fields can be endowed with derivations that mimic the derivation of germs of real functions in a Hardy field [Sch01,vdDMM01,KM11]. We propose the following unifying notion for transseries and Exp-Log series, which we believe applies to the field of surreal numbers:

Definition 5.1 LetK=R((Γ)) be a field of generalized series endowed with a partial logarithm log :Γ→R((Γ^≻1)). A complete subfieldL⊂Kwhich containsΓis called a field of exp-log-transseries if the following properties hold:

TEL1. domain log=L_>0. TEL2. log(Γ)=L^≻1. TEL3. log(1+ǫ)=

∞

X

n=1

ǫⁿ/n∈L^≺1for anyǫ∈L^≺1.

TEL4. For any sequence of monomials (mn)n ⊂ Γsuch that for any n ∈ N, mn+1 ∈ Supp log(mn), then there exists a rank N ∈Nsuch that log(mN+k)=mN+k+1for any k∈N.

Any field of exp-log-transseries is both a field of transseries (our four axioms are specializations of the four axioms of [Sch01, Definition 2.2.1]) and a field of exp-log series. Indeed, denote byΦ⊂Γthe biggest subset of monomials stable by log (i.e. the set of log-atomic elements in [Sch01, Section 2.2]). Then resuming the notations and terminology of [KM11],Φis a totally ordered set of fundamental monomials. For the corresponding Hahn series field, we setL₀ =R((H(Φ)))⊂Lwhere H(Φ) denotes the Hahn group overΦ. Then by (TL4) one has thatL=L^EL₀ .

We believe that:

Conjecture 5.2 The field of surreal numbers is an exp-log-transseries field. The sur- realsκ_a,nfor a ∈N⋊, n∈Z}, are the log-atomic elements as well as the initial funda- mental monomials.

In particular, this would allow us to use the results of [Sch01,KM11] to endow the surreal numbers with Hardy type derivations.

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